Invariants
Base field: | $\F_{5^{4}}$ |
Dimension: | $2$ |
L-polynomial: | $( 1 - 49 x + 625 x^{2} )( 1 - 48 x + 625 x^{2} )$ |
$1 - 97 x + 3602 x^{2} - 60625 x^{3} + 390625 x^{4}$ | |
Frobenius angles: | $\pm0.0637685608585$, $\pm0.0903344706017$ |
Angle rank: | $2$ (numerical) |
This isogeny class is not simple, primitive, ordinary, and not supersingular. It is principally polarizable.
Newton polygon
This isogeny class is ordinary.
$p$-rank: | $2$ |
Slopes: | $[0, 0, 1, 1]$ |
Point counts
Point counts of the abelian variety
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $333506$ | $151728554700$ | $59593325970144968$ | $23282931123531240192000$ | $9094945732251198388598540066$ |
Point counts of the (virtual) curve
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $529$ | $388421$ | $244094260$ | $152587017409$ | $95367418161409$ | $59604644663623586$ | $37252902987502815505$ | $23283064365593187091969$ | $14551915228374957999597364$ | $9094947017729544342083496101$ |
Jacobians and polarizations
This isogeny class is principally polarizable, but does not contain a Jacobian.
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{5^{4}}$.
Endomorphism algebra over $\F_{5^{4}}$The isogeny class factors as 1.625.abx $\times$ 1.625.abw and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is: |
Base change
This is a primitive isogeny class.